关于完全链代数的拟同构型
On the quasi-isomorphism type of a perfect chain algebra
摘要：
令$R$为（P.I.D），令$T(V),\partial)$为自由$R$-dga。$(T(V),\partial)$的准同构类型是所有与$(T(V),\partial)$准同构的自由dgas的集合，记为$\{(T(V),\partial)\}$。本文研究了一类新的自由dga的集合$\{(T(V),\partial)\}$的刻画和计算问题（一类特殊的完全dga是使得$H_{*}(X,R)$是自由的单连通CW-复形的Adams-Hilton模型）.证明了若$(T(V),\partial)$和$(T(W),δ)$是两个完全数据集，则$(T(W),δ)\in \{(T(V),\partial)\}$当且仅当它们的Whitehead正合列同构.证明了每个dga $(T(V),\partial)$都可以分裂成一对$\big((T(V),\widetilde{\partial}),(π_{n})_{n\geq 2}\big)$，这对$\big((T(V),\widetilde{\partial}),(π_{n})_{n\geq 2}\big)$由一个完美的dga $(T(V),\widetilde{\partial})$和一个扩展族$(π_{n})_{n\geq 2}$组成，并证明了如果$(T(W),\widetildeδ)\in \{(T(V),\widetilde{\partial})\}$和扩展族$(π_{n})_{n\geq 2}$与$(π'_{n})_{n\geq 2}$同构（在某种意义上），则$(T(W),δ)\in \{(T(V),\partial)\}$ .
英文摘要：
Let $R$ be a (P.I.D) and let $T(V),\partial)$ be a free $R$-dga. The quasi-isomorphism type of $(T(V),\partial)$ is the set, denoted $\{(T(V),\partial)\}$, of all free dgas which are quasi-isomorphic to $(T(V),\partial)$. In this paper we investigate to characterize and to compute the set $\{(T(V),\partial)\}$ for a new class of free dgas called perfect (a special kind of a perfect dga is the Adams-Hilton model of simply connected CW-complex such that $H_{*}(X,R)$ is free). We show that if $(T(V),\partial)$ and $(T(W),δ)$ are two perfect dgas, then $(T(W),δ)\in \{(T(V),\partial)\}$ if and only if their Whitehead exact sequences are isomorphic. Moreover we show that every dga $(T(V),\partial)$ can be split to give a pair $\big((T(V),\widetilde{\partial}),(π_{n})_{n\geq 2}\big)$ consisting with a perfect dga $(T(V),\widetilde{\partial})$ and a family of extensions $(π_{n})_{n\geq 2}$ and we establish that if $(T(W),\widetildeδ)\in \{(T(V),\widetilde{\partial})\}$ and if the extensions $(π_{n})_{n\geq 2}$ and $(π'_{n})_{n\geq 2}$ are isomorphic (in a certain sense), then $(T(W),δ)\in \{(T(V),\partial)\}$.
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